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Consider a linear system of *n* equations in *n*
unknowns, written in compact form: *A x =
b*. If the matrix

A consequence of this observation is an important theorem, known as Cramer's rule, that, though not much used in practice, is very important from a theoretical point of view because it reveals explicitly how the solution depends on the coefficients of the augmented matrix.

Given a linear system of *n* equations in *n*
unknowns, call Δ the determinant of the matrix *A*
of the system, Δ_{i} the determinant of the matrix
formed by replacing the *i*-th column of the matrix
*A* by the constants *b _{1}, b_{2},
...,b_{n}*. Then if ≠0 the unique solution of the
system is

.

Solve the system *A x =
b* where
and . It's easy to find that
det(

.

We can easily check out the solution. We have: . So .

copyright 2000 et seq. maddalena falanga & luciano battaia

first published on march 15 2002 - last updated on september 01
2003